p-capture reaction cycles in rotating massive stars and their impact on elemental abundances in Globular Cluster stars: A case study of O, Na and Al

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1 Research in Astron. Astrophys. Vol.0 (200x) No.0, Research in Astronomy and Astrophysics p-capture reaction cycles in rotating massive stars and their impact on elemental abundances in Globular Cluster stars: A case study of O, Na and Al Upakul Mahanta 1, Aruna Goswami 2, H. L. Duorah 3 and Kalpana Duorah 4 1 Department of Physics, Bajali College, Pathsala, PIN ,India; upakulmahanta@gmail.com 2 Indian Institute of Astrophysics, Bengaluru, PIN ,India; aruna@iiap.res.in 3 Department of Physics, Gauhati University, Guwahati, PIN ,India; hl duorah@yahoo.com 4 Department of Physics, Gauhati University, Guwahati, PIN ,India; khowang56@yahoo.co.in Received 2016 August 25; accepted 2016 April 21 Abstract Elemental abundance patterns of Globular Cluster stars can provide important clues for understanding cluster formation and early chemical evolution. The origin of the abundance patterns, however, still remains poorly understood. We have studied the impact of p-capture reaction cycles on the abundances of oxygen, sodium and aluminium considering nuclear reaction cycles of Carbon-Nitrogen-Oxygen-Fluorine, Neon-Sodium and Magnesium-Aluminium in massive stars in stellar conditions of temperature range, to K and typical density of10 2 gm/cc. We have estimated abundances of oxygen, sodium and aluminium with respect to Fe, which are then assumed to be ejected from those stars because of rotation, rotating at the critical limit. These ejected abundance of elements are then compared with their counterparts what has been observed in some metal-poor evolved stars, mainly giants and red-giants, of globular clusters M3, M4, M13 and NGC6752. We observe an excellent agreement with [O/Fe] between the estimated and observed abundance values for globular clusters M3 and M4 with a correlation coefficient above 0.9 and a strong linear correlation for the rest of two clusters with a correlation coefficient above 0.7. The estimated [Na/Fe] is found to have a correlation coefficient above 0.7 and thus implying a strong correlation for all the four globular clusters. As far as [Al/Fe] is concerned it also shows a strong correlation between the estimated abundance and the observed abundance for globular clusters M13 and NGC6752, since here also the correlation coefficient is above 0.7 whereas for globular cluster M4 there is a moderate correlation found with a correlation coefficient above 0.6. Possible sources of these discrepancies are discussed. Key words: Galaxy: globular cluster: indivisual (M3, M4, M13, NGC6752) - stars: abundances - stars: massive 1 INTRODUCTION Globular cluster (GC) stars are known to display homogeneous abundance patterns for Fe-peak elements but significant abundance variations are seen among the light elements. Na-O and Mg- Al anticorrelations are also observed among the stars of GCs (Kraft (1997), Gratton et al. (2004), Carretta et al. (2009), Smolinski et al. (2011)) etc and references therein). The inhomogeneities and

2 2 Upakul Mahanta et al. abundance variations observed in light-elements increases with decreasing metallicity. For instance, C, N, O and Na abundances in giants of M7 with an average metallicity ([Fe/H]) of 0.7 show mild variations compared to their counterparts observed in giants of M5 with a metallicity of 1.2. Several studies both observational and theoretical in literature are devoted to the understanding of the complex abundance patterns of GCs (Cordero et al. (2015), Roederer et al. (2015), Spite et al. (2015), Villanova et al. (2016)). The origin of the observed abundance anomalies, however,still remain poorly understood. GCs span a wide range in metallicities with [Fe/H] as small as 2.38 upto as large as (Table 2 (Gratton et al. (2004)) and references therein). Here amongst the chosen GCs, GC M3 has an average metallicity of 1.39 and is more distant to M5. The scatter among the light elements and the enhanced odd atomic numbered elements typically seen in metal-poor GCs are also noticed in M3, although at a lower level of star-to-star variation (Cohen et al. (2005)). GC M13 with an average metallicity 1.50 is known to show a large star-to-star differences in the abundance of Al, Mg, Na, and O among its red giants (Cohen et al. (2005) references therein).gc M4 is the closest to us (Dixon et al. (1993)) and has an average cluster metallicity of 1.17 (Liu et al. (1990), Drake et al. (1992), Drake et al. (1994)). A characteristic feature observed in the colour-magnitude diagrams of GC M4 stars is the presence of broadened red-giant branches (Marino et al. (2008)). The two giant branches appear to correlate with typical globular cluster variations in [O/Fe] and [Na/Fe] abundances rather than variations in total [C+N+O/Fe] abundance (Martell et al. (2011)). Detailed elemental abundances are available for a large number of stars belonging to this cluster. The largest spread in light element abundances is known to be observed in GC NGC This GC has been studied by many groups including (Yong et al. (2005)), and the reported average cluster metallicity due to (Yong et al. (2005)) is But the common feature of these four GCs is the observed Na-O and Mg-Al anti-correlation. The proton-capture chains that convert C and O to N, Ne to Na and Mg into Al in the hydrogen-burning layers of evolved stars are believed to be responsible for these observed trends; however, the astrophysical site(s) for their occurrence is still under debate. Amongst the contesting scenarios one is evolutionary which considers deep mixing of the stellar envelope through the hydrogen burning shell that brings the products of proton-capture chains to the surface. And the other one is primordial where observed abundances originated from proton-capture synthesis that took place in some earlier generation massive stars whose interior can easily attain a high temperature so that advanced H-burning can be sustained. However recent spectroscopic studies have challenged the former one as it possibly can t explain why a similar kind of abundance anomalies is present even in stars which are below and above the main sequence turn-off (Decressin et al. (2007b)). Also the evolutionary scenario can t explain why the stars of GC ω Cen which is the most massive cluster, have found with a large spread in metallicity and thereby throwing out the earlier paradigm that the GCs are examples of Simple Stellar Populations (SSP) (D Ercole et al. (2008)). Under the currently favoured astrophysical sites where the processing of element takes place, as well as the mechanism by which processed material is subsequently delivered into new generations of stars are (i)the massive AGB stars (D Ercole et al. (2008)), (ii) the fast-rotating massive main sequence stars (Decressin et al. (2007b)) and (iii) the massive interacting binaries (de Mink et al. (2009)). In AGB star during Hot Bottom Burning (HBB), the CNO elements undergo p-capture nucleosynthesis which is found to be sensitive to metallicity of the star (Ventura et al. (2013)). But this HBB, for any effective depletion of oxygen abundance due to p-capture reaction, requires a high temperature as high ast 9 > 0.1 (Ventura et al. (2013)). Again at high temperature the destruction channel for sodium is dominant compared to the production reaction by p-capture on 22 Ne nuclei, thus the sodium previously accumulated at the surface is destroyed (Denissenkov & Herwig (2003), Ventura et al. (2013)). This points towards correlation between Na and O abundance which in fact has been confirmed by a recent study performed by (Ventura et al. (2013)) at very low metallicity unless the polluted material which is lost from the surface of the stars via slow winds suffer a certain amount of dilution with pristine (Ventura et al. (2013)) to have Na-O anticorrelation, which is then remain trapped within the gravitational potential of the cluster. As mentioned earlier, these four GCs have shown Na-O anticorrelation, three of which has been already reported in a recent study performed by (D Antona & Ventura (2007)) considering some new stellar models. Again in the third scenario i.e. interacting massive binary stars may provide an efficient way to lose large

3 p-capture reaction cycles (I) 3 amounts of processed material that could be incorporated into an enriched population. This scenario is attractive because a large fraction of massive stars are indeed observed to be members of binaries that will interact during their lifetime (Sana et al. (2013)). Again de Mink et al. (2009) has assumed a non-conservative evolution of massive close binaries which may not be correct because the mass that is lost by the loser (the primary one) leaves the binary as a slow wind driven by the rotation of the gainer (the secondary one) and takes with the specific orbital angular momentum of the gainer. Thus this will lead to the merger of the binary and the evolution will be different as suggested by the same author (Vanbeveren et al. (2012)). In the winds from fast rotating massive stars (WFRMS) scenario (discussed in section 2), it is assumed that massive stars within GCs rotate near break-up speed. Processed material is brought to the surface by rotational mixing, lost via a mechanical wind, and then accumulates in a disk around the star, where the second generation of (low-mass) stars is assumed to form (Decressin et al. (2010)). In this work we have considered the nuclear burning cycles Carbon-Nitrogen-Oxygen-Fluorine (CNOF), Neon-Sodium (NeNa), and Magnesium-Aluminium (MgAl) in rotating massive star at high temperature and low density conditions and have estimated the abundances of the product elements O, Na and Al in a temperature range of to K since CNO, NeNa and MgAl cycles activated above temperature K, K and K respectively (Decressin et al. (2007b)). Although nucleosynthesis during NeNa and MgAl cycles are discussed at length in (Arnould et al. (1999)) and (Jose et al. (1999)),the availability of new estimates of the cross-sections for many of the reactions involved in these cycles, from recent experimental determination (Iliadis et al. (2010)), prompted us to reinvestigate the synthesis of elements due to these cycles. Our aim is to compute the abundances of the key elements, O, Na, and Al which are important diagnostics for understanding the chemical evolution, and thereby examine the impact of proton-capture reactions on the observed abundances of GC stars. A comparison between the computed and observed data suggests contributions coming from other sources, yet to be identified, that might have possibly influenced the observed abundances. In Section 2 we present the physical conditions for the occurrence of the nuclear burning cycles and lifetime measurements for proton-capture reactions. The evolution of the stable elements in the cycles at equilibrium condition due to hydrogen burning is presented in Section3. Section4deals with CNOF and other nuclear burning cycles and the abundance calculation of the selected elements. Discussion and Concluding remarks are presented in Section 5. 2 THE PHYSICAL SITUATION Classical models of stellar evolution focus on the dominant role of various stages of nuclear burning in the stellar core. Not only the temperature is crucial in any stellar situations but density also plays an important role. Even though the density ranges from many orders of magnitude its response in the burning rates is only linear and hence its importance is much less significant as far as the elemental synthesis is concerned unless one considers a situation where low temperature and high density is prevailing. The materials are synthesized in the inner regions near to the core and are brought to the surface by means of some physical mechanisms like convection or by dredging up mechanisms. Here the stars are assumed to be spherically symmetrical with no magnetic field, no rotation and no mass loss from the surface. And the process of convection is only the mixing mechanism with the convective regions and are always fully mixed (Salaris et al. (2002)) which is the canonical model of star. But in recent years it has become clear that stellar evolution, particularly for more massive stars, can also be profoundly influenced by the loss of mass and angular momentum from the stellar envelope and surface (Maeder (2008)). Models of the MS evolution of rotating massive stars show that, at the surface, the velocity approaches a critical limit. This is induced by internal evolution of stars that results in transport of angular momentum from the contracting, faster rotating inner convective core to the expanding, slowed down radiative envelope (Meynet et al. (2006)). However beyond the critical limit any further increase in rotation rate is not dynamically allowed, hence the further contraction of the interior is then balanced by a net loss of angular momentum through an induced mass loss. Here this mass loss is assumed to be isotropic. Thus the sythesized mass by the star will now be delivered to the ISM. And

4 4 Upakul Mahanta et al. hence the ISM is pre-enriched with material those have produced by other stars. Moreover the materials are assumed to be released in the ISM with a very low velocity and thus can easily be retained into the potential well of the globular cluster. Here we assume the density of the stellar matter constant which in general, is not a sensitive parameter unless one goes to a high value 10 5 gm/cc or even higher where pycnonuclear type of reaction has to be considered. We also assume the star to be in equilibrium and thereby making it to maintain a balance between the produced energy due to thermonuclear fusion and the loss of energy at the surface. Thus the temperature of the star can essentially assume to be a fixed quantity. Here we mainly focus on which type of stars did produce the material enriched in H-burning products and how? However we do not comment on the physical factors that influence the mass loss neither we look for what is the physical mechanism responsible for selecting only material bearing the signatures of H-processing those have been lost? 3 HYDROGEN BURNING NUCLEOSYNTHESIS The rate of nuclear reactions is dependent on the density of the reactants, the velocity of one reactant relative to another, and the probability of a reaction occurring. Mathematically R = 1 1+δ ij N i N j < σv > (1) where i,j are two separate species and δ ij is the function preventing the double counting of those two species. For computational simplification the number densityn i s of any element with mass number A i can be expressed in terms of its mass fraction X i by N i = ρx i N A /A. Here ρ is the density, N A is Avogadro s number. The thermonuclear reaction rate for a proton-capture reaction then takes the form R = ρ2 N A A H A i [X H X i (N A < σv >)] cm 3 sec 1 (2) Here, N A < σv > is the reaction rate constant and X i is the mass fraction of any other heavy element. From this the lifetime against proton-capture for the elements in the enhanced state is given by the following equation 1 τ p = sec (3) ρx H [N A < σv >] 4 CNOF CYCLE If heavier elements are present in some stellar condition where the temperature and the density is low, then 12 C(p,γ) 13 N reaction can compete with p-p reaction and thereby can initiate the CNO burning mechanism (Clayton (1983) and ref. in here). 13 N is β unstable and decays to 13 C in a time scale of sec (Audi et al. (2003)) since its proton capture lifetime is quite large in the considered temperature density condition. On the other hand 13 C is a stable isotope of carbon with relative abundance (Lodders (2003)). This 13 C forms 14 N by taking a proton and then 14 N to 15 O taking a proton again. 15 O then decays via aβ emission to 15 N in an average lifetime of sec (Audi et al. (2003)). Here 15 N branching appears. The table 1 shows the branching ratios of the cycle at various temperatures. These branching ratios show that the (p,α) reaction wins over the (p,γ) reaction and thus confirming the cyclic behaviour forming 12 C by most of 15 N nuclei. This is the CN cycling. The branching ratio will guide how much of 15 N will go to form 16 O by taking a proton. This point onwards the oxygen-fluorine reaction network starts,of which discussion is available in many places in the literature (Bethe (1997), Cameron et al. (2013)). Here we give a brief description relevant to this work. A proton capture by O 16 leads to the formation of unstable F 17 which decays into 17 O. This secondary isotope of oxygen has very small relative abundance (Lodders (2003)). This is probably due to the fact that O 17 gets destroyed via both O 17 (p,α)n 14 and O 17 (p,γ)f 18 ; the former reaction rate is higher compared to the later one at the same temperature condition. Thus the (p,α) reaction also has its importance for further evolution and starts competing with (p,γ) reactions. The inclusion of the (p,α) reaction

5 p-capture reaction cycles (I) 5 which has also been considered here, introduces another branching point in the cycle, of which the branching point ratio of (p,γ) to (p,α) which has been shown in (Table 1). If the cycle advances forming 18 F then this will be immediately followed by F 18 (e +,ν e )O 18 reaction. This is because 18 F which is unstable against β-decay and at low density such as ρ 2 (= ρ/10 2 g/cm 3 )= 1 and, for temperature 0.02 T 9 0.1, the p-capture lifetime of 18 F is very large as compared to its β decay mean lifetime ie 9504 sec (Audi et al. (2003)). 18 O the tertiary isotope of oxygen has very low abundance compared to 16 O but greater as compared to 17 O (Lodders (2003)).This point onwards we take the reaction steps given in (Hansen et al.(2004), Mountford (2013)) ie 18 O(p,γ) 19 F. Sometime the radiative capture on 18 O can t be neglected as compared too 18 (p,α)n 15 even though the (p,α) channel is substantially stronger then (p,γ) channels. Because depending upon the spin and energy of resonance the latter can be of comparable strength (Weischer et al. (1980)). Still we have checked for possible alteration of abundances of oxygen by including of a third branching point at 18 O. But still it doesn t lead us to find any significant change in the mass fraction of 16 O which has been found to be consistent with the earlier report of Audouze et al. (1977) where the author had mentioned that the 18 O(p,γ) 19 F leak has little effect on CNO equilibrium abundance. For instance the 16 O abundance changes by 12% only, at temperatures T 9 =0.03 and0.05 (Table5). Moreover the goal of this choice is because it may improve our knowledge of levels in the 19 F nucleus that are relevant to nuclear astrophysics and hopefully for a possible fluorine production network. Recently Buckner et al. (2012) had studied 18 O(p,γ) 19 F reaction and have found that most 19 F levels decay by γγ-cascades through the first (110 kev) excited state, and all 19 F levels (with known decay schemes) de-excite through the second (197 kev) excited state. Moreover it is an interesting element in the periodic table because of the fact that though it is surrounded by some of the most abundant elements in the universe like oxygen, nitrogen and neon, yet it is itself very rare. Perhaps it is because an odd Z element with only one single stable isotope and it is very fragile with its 9 protons and 10 neutrons (Palacois (2006)). Then the finally produced 19 F which is destroyed by a (p,α) reaction forming 16 O since 19 F(p,α) 16 O (p,α) reaction rate is faster as compared to (p,γ) reaction which would have produced 20 Ne.Thus the CNOF cycle is 12 C +p 13 N +γ 13 N 13 C +e + +ν e 13 C +p 14 N +γ 14 N +p 15 O +γ 15 O 15 N +e + +ν e 15 N +p 12 C +α 15 N +p 16 O +γ 16 O +p 17 F +γ 17 F 17 O +e + +ν e 17 O +p 14 N +α 17 O +p 18 F +γ 18 F 18 O +e + +ν e 18 O +p 19 F +γ 19 F +p 16 O +α Table 1 The branching ratio(b r = N A <σv>p,γ N A <σv>p,α ) at various temperature in units oft9. The rate constants for 15 N are taken from NACRE compilation and for 17 O are taken from (Iliadis et al. (2010)) T 9 15 N(B r) 17 O(B r)

6 6 Upakul Mahanta et al. To calculate the lifetimes of all the reactions at the density value, the required reaction rate constants are taken from (Iliadis et al. (2010)) which are the recommended medium rate constant values except for the (p,α) reactions. The reaction rate constants for the (p,α) reactions reaction, involved in this cycle have been taken from NACRE compilation. Estimated proton-capture lifetimes for various elements are listed in Table 2 which are relevent to our reaction cycle. Table 2 Estimated proton-capture reaction lifetimes for the nuclear reactions. The rate constants ie N A < σv > are in cm 3 mol 1 sec 1, T 9 is in the unit of 10 9 K and ρ 2 is in the unit of 10 2 gm cc 1.The τ β for 13 N, 15 O, 17 F and 18 F are , , and 9504 sec respectively. The τ β and the τ p represent the β-decay and the proton-capture lifetimes respectively in sec. Reaction T 9 N A < σv > τ p(ρ 2 = 1) C(p,γ) 13 N C(p,γ) 14 N N(p,γ) 14 O N(p,γ) 15 O N(p,γ) 16 O O(p,γ) 16 F O(p,γ) 17 F F(p,γ) 18 Ne O(p,γ) 18 F

7 p-capture reaction cycles (I) 7 Reaction T 9 N A < σv > τ p(ρ 2 = 1) F(p,γ) 19 Ne O(p,γ) 19 F F(p,α) 16 O NeNa cycle: At high temperature conditions additional hydrogen burning cycles may come in the form of NeNa cycle (Marion(1957)). Because of the high coulomb barrier they may not be seem important from the energy sources point of view. However, the enhancement of Na which has been observationally found in many red giant and supergiant stars demands the investigation of NeNa cycle as a possible candidate of Na enhancement. Typically this cycle starts fromne 20 which capture a proton to formna 21. This unstable Na 21 forms Ne 21 via a β emission. However the reaction Ne 21 (α,n)mg 24 can be a source of neutron (Burbidge et al. (1957)) ifαparticles are available but the branching ratio ( NA<σv>α,n N A<σv> p,γ )<<1 guarantees the production ofna 22 throughne 21 (p,γ)na 22 which is a relatively long-lived isotope of Na with an average lifetime of sec. Na 22 produces Ne 22 via Na 22 (e +,ν e )Ne 22. We have also taken into account of the reaction Na 22 (p,γ)mg 23 that may affect the energy production rate, cycle lifetime and the abundance of Na. Other β-decay life times can not compete with the proton-capture lifetime so long as the value of temperature T Production of Mg 25 from Ne 22 via (α,n) reaction is ignored as the temperature required (T 9 = 0.3) for this reaction to occur is beyond the temperature range we have considered. The reactions in the NeNa cycle will depend upon the competition between proton capture and β-decay lifetimes which again in turn depend upon the density and temperature condition. These factors are taken into account, and thus the NeNa cycle as below: 20 Ne+p 21 Na+γ 21 Na 21 Ne+e + +ν e 21 Ne+p 22 Na+γ 22 Na 22 Ne+e + +ν e 22 Ne+p 23 Na+γ 23 Na+p 20 Ne+α Thus Ne 20 can be formed when Na 23 captures a proton as the nuclear reaction rates for Na 23 (p,α)ne 20 is higher ( ) (Iliadis et al. (2010)) compared to the competing reaction Na 23 (p,γ)mg 24 which is sufficient to guarantee NeNa cycling. In Table 3 we present the protoncapture lifetimes for various elements at considered stellar condition. The reaction rate constants are taken from (Iliadis et al. (2010)). MgAl cycle: This cycle, initiated by Mg 24 with a proton-capture leads to the formation of unstable Al 25. Mg 24 is the most abundant element of the cycle for temperature T because of the slow reaction rate of Mg 24 (p,γ)al 25. Al 25 quickly disintegrates to Mg 25 in an average time scale of sec (Audi et al. (2003)).Mg 25 through a (p,γ) reaction forms the other isotopes of Al. The first one isal 26 which exists in two states, a ground state Al 26g and an isomeric state Al 26m. If temperature T 9 1, bothal 26g (T β = sec; Audi et al. (2003)) andal 26m (T β = sec; Audi et al. (2003))

8 8 Upakul Mahanta et al. Table 3 Estimated proton-capture reaction lifetimes for the nuclear reactions. The rate constants ie N A < σv > are in cm 3 mol 1 sec 1, T 9 is in the unit of 10 9 K and ρ 2 is in the unit of 10 2 gm cc 1. The τ β for 21 Na and 22 Na are and sec respectively. Theτ p represents the β-decay and the proton-capture lifetimes respectively in sec. Reaction T 9 N A < σv > τ p(ρ 2 = 1) Ne(p,γ) 21 Na Na(p,γ) 22 Mg Ne(p,γ) 22 Na Na(p,γ) 23 Mg Ne(p,γ) 23 Na Na(p,α) 20 Ne quickly attains equilibrium. At T (which falls within the temperature range considered here) the equilibrium does not get established and hence both the species have to be treated separately (Ward et al. (1980)). Thus, if the temperature T 9 < 0.03, destruction of Al 26g mainly occurs through a β-decay reaction at both the density values considered here. Assuming this isotope of Al to be stable we have considered Al 26g (p,γ)si 27 (e +,ν e )Al 27 at the considered range of temperatures. The destruction of Al 27 again depends upon the stellar temperature. As long as the temperature T 9 remains less than 0.08 the (p,α) reaction wins over (p,γ) reaction (Iliadis et al. (2010)) and that is sufficient to confirm the cyclic nature of Mg-Al burning modes. But if T 9 goes beyond that value there will be leakage of Al 27 through Al 27 (p,γ)si 28 which is being ignored in the present case. For the MgAl cycle the following reaction chains are being considered. 24 Mg +p 25 Al+γ 25 Al 25 Mg +e + +ν e 25 Mg +p 26g Al+γ 26g Al+p 27 Si+γ 27 Si 27 Al+e + +ν e 27 Al+p 24 Mg +α The proton-capture lifetimes have been calculated for the cycle using the recommended reaction rate constants from (Iliadis et al. (2010)). Once again the nuclear reaction sequence in the MgAl cycle depends on the competition between the proton-capture and β-decay lifetimes. Estimated proton-capture lifetimes for various elements are presented in Table 4.

9 p-capture reaction cycles (I) 9 Table 4 Estimated proton-capture reaction lifetimes for the nuclear reactions. The rate constants ie N A < σv > are in cm 3 mol 1 sec 1, T 9 is in the unit of 10 9 K and ρ 2 is in the unit of 10 2 gm cc 1. The τ β for 25 Al, 26g Al and 27 Si are , and sec respectively. The τ p represents the β-decay and the proton-capture lifetimes respectively in sec. Reaction T 9 N A < σv > τ p(ρ 2 = 1) Mg(p,γ) 25 Al Al(p,γ) 26 Si Mg(p,γ) 26g Al g Al(p,γ) 27 Si Mg(p,γ) 27 Al Si(p,γ) 28 P Al(p,α) 24 Mg Evolution of Elemental abundances In the reaction cycle considered, the cycle produces oneα-particle along with twoν e s and twoe + s. The initial nuclei act mainly as catalysts. Although there is always a consumption of hydrogen the total mass and the total number of nuclei in each cycle remain conserved. Thus the net effect for all the three cycles is 4H 1 He 4 +2e + +2ν e +γ, dn H dt < 0, dn He dt > 0 (4) The generalised differential equation that governs the evolution of any element in terms of number density via a proton-capture reaction or a β-decay or both at the enhanced condition is given by (Clayton (1983)) dn i dt = N i N H < σv > p,i +N j N H < σv > H,j ±λ k N k (5) where λ k is the decay constant of an unstable nucleus with number density N k. As all the three cycles involved proton-capture reactions and β-decays, the abundances will primarily depend upon the lifetime

10 10 Upakul Mahanta et al. of these processes. If the β-decay lifetimes, τ β for an unstable element in the cycle is shorter than the proton-capture lifetime τ p for the same element then the β-decay lifetimes can be bypassed and thus the element can be thought of representing the next stable element in the cycle having the same mass number. Considering that the equation(5) for each cycle takes the form of the following differential rate equations in terms of mass fraction of any element dx i dt = ( R p,i X i Xρ+ A ) i R p,j X j Xρ A j where R p,i s are [N A < σv >] terms for the respective proton-capture reactions. A i and A j stand for mass number of different nuclei. Equation (6) can be expressed as a function of the hydrogen mass fraction to get a series of first order simultaneous linear differential equations for each cycle as which are then solved for suitable initial condition. 4.2 Calculation of abundances ( ) dx R p,i X i + Ai i A j R p,j X j = [ dx H ( )] (7) A j 1 A i A i R p,i X i In the case of CNOF cycle the initial abundance of heavy elements have been chosen in such a manner that it does not affect the production of primary nitrogen from being the highest abundant element in the CNO reaction, which actually gets hampered for metallicities higher than Z= Moreover at higher metallicities the rotational mixing is not so efficient. Neither we take very or extremely low metallicities because low metallicity reveals a low content of heavy elements and thereby making the opacity lower. Hence the stars become more compact, and therefore hotter so they are going to have high luminosity. This high luminosity increases the radiation pressure and thus stellar mass loss (Meynet et al. (2009)). Again in the case of fast rotating massive stars, with an initial metallicity Z= , with typical time averaged velocity of 500 kms 1 on the main sequence easily reach the critical velocity (Decressin et al. (2007b)) at the beginning of their evolution and remain near the critical limit during the rest part of their main sequence. As a consequence, they lose large amount of material through a mechanical wind, which probably leads to the formation of a slow outflowing Keplerian equatorial disk. Keeping in view of all these the simultaneous linear first order differential equations are solved numerically for each cycle varying the hydrogen mass fraction up to X = 0.60 to get the equilibrium mass fraction abundances of the stable heavy elements with the initial condition as X = 0.70, Y = 0.298, such that X + Y + Z = 1 where Z is the sum of the mass fraction of elements C 12,N 14,O 16,and F 19 distributed equally ie Z C 12 = Z N 14 = Z O 16 = Z F 19 = for CNOF then X = 0.70, Y = , Z Ne 20 = for NeNa and lastly X = 0.70, Y = , Z Mg 24 = for MgAl cycle (Meynet et al. (2008)). The equilibrium abundance by mass fraction of the stable isotopes of considered elements are taken up to the first decimal place. We have also calculated the range of uncertainty in the equilibrium mass fraction abundance of O 16, Na 23 and Al 27 due to the uncertainty in the reaction rate constants considering both low and high rate constants for respective reactions (Iliadis et al. (2010)) as well as NACRE compilation. These are presented in Table 5. These abundance by mass fraction of the stable isotopes of any element (x) with respect to Fe are then calculated using the expression Here [ x H] [ x ] [ x = Fe H] [ ] Fe H [ ] [ ] Nx Nx = log log N H star N H (6) (8)

11 p-capture reaction cycles (I) 11 Table 5 The estimated abundance by mass fraction of O 16, Na 23 and Al 27 due to the uncertainty in the reaction rate constants with low, medium (recommended) and high reaction rate constant. T 9 O 16 Na 23 Al 27 Low Medium High Low Medium High Low Medium High **** ***** ***** **** ***** ***** Table 6 The estimated abundance ratios of [O/Fe] and [Na/Fe] for GC M3. a = (Cohen et al. (2005)), b = this work. [ star Fe ] a [ O ] a [ Na a [ O ] b [ Na ] b H Fe Fe] Fe Fe VZ ± ± II ± ± VZ ± ± III ± ± IV ± ± C ± ± IV ± ± III ± ± III ± ± C ± ± V < 0.03± ± V ± ± C ± ± and [ ] Fe = log H [ NFe N H ] star log [ NFe The quantities N x and N H are the solar number densities taken from (Lodders (2003)). For oxygen we have taken the equilibrium O 16 mass fraction for the abundance calculations since its relative abundance is larger compared to the other two stable isotopes of oxygen. The calculated abundances of O, Na and Al are presented in Tables 6, 7, 8 and 9,for density ρ 2 = 1 for the clusters M3, M4, M13 and NGC6752 respectively. The observed abundances of a few metal poor evolved stars from (Cohen et al. (2005), Ivans et al. (1999), Cohen et al. (2005) and Yong et al. (2005)) are also presented in these tables for a comparison. The calculated abundances at a particular temperature are those for which the difference between the observed and the calculated abundance is found to be minimum. Here most of the stars have gotten impacted since the influence of the wind is necessary in order to explain the GC abundance anomalies Decressin et al. (2007b). Moreover, the second generation of stars are formed because of the ejection of material from the rapidly rotating massive stars which are mixed with primordial material left over from the star formation process (Bastian et al. (2014)). A recent study conducted by Bekki (2011) has shown that in massive star clusters second generation of stars are formed from the gaseous ejecta of AGB stars if the mass of cluster exceeds 10 6 M. Thus this gives a possibility of the presence of many massive stars within the same GC. A comparison between the computed and observed abundances from (Cohen et al. (2005), Ivans et al. (1999), Cohen et al. (2005) and Yong et al. (2005)) is shown in Fig. 3, 4, 5 and 6 respectively. In Fig.6 the observed [O/Fe] and [Na/Fe] values are with σ <0.1 dex (Yong et al. (2005)). N H ]

12 12 Upakul Mahanta et al. Table 7 The estimated abundance ratios of [O/Fe],[Na/Fe] and [Al/Fe] for GC M4. a = (Ivans et al. (1999)), b = this work. [ star Fe ] a [ O ] a [ Na ] a [ Al a [ O ] b [ Na ] b [ Al ] b H Fe Fe Fe] Fe Fe Fe L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± L ± ± ± DISCUSSION AND CONCLUDING REMARKS Synthesis due to the proton-capture reaction has been investigated using updated reaction rates (Iliadis et al. (2010)) for nuclear reaction cycles CNOF, NeNa, and MgAl under conditions of high temperature and low density. Abundance ratios, i.e., [O/Fe], [Na/Fe] and [Al/Fe] are also calculated and compared with their counterparts observed in a sample of metal poor stars of GC M3, M4, M13 and NGC6751 (Figures 2, 3, 4 and 5). This comparison demonstrates the extent to which these nuclear reaction cycles can explain the observed abundances of O, Na and Al. These ratios show similar trends for stars of all the four GCs considered here. We find that within the temperature and density conditions considered, there exists a small range of temperature values at which the computed [O/Fe], [Na/Fe] and [Al/Fe] ratios agree well with the observed ratios in a sample of stars of GCs M3, M4, M13 and NGC6752. GC stars are known to show Na-O anti-correlation. This anti-correlation is a common feature for globular clusters in the range of metallicity ( 2.5 [F e/h] 1). To examine this effect, we have plotted the computed [Na/Fe] vs [O/Fe] and compared with the observations (Figure 6). The computed abundance ratios are found to follow similar trends for all the GCs. We also have produced the Na-O anti-correlation range 0.80 [N a/o] 0.40 collectively for all the four clusters, which is, well in the comparable range observed 0.62 [N a/o] 1 (Decressin et al. (2007)). Probably a more accurate result can be obtained if one considers the 23 Na(p,α) 20 Ne reaction rate lowered by a factor2 for none solar-scaled metallicity (Ventura& Antonna (2006)). Moreover recently Buckner et al. (2012) has reported an updated set of 18 O(p,γ) 19 F reaction rate constants which may again give even more fine tuned results. We have also found the Mg 24 mass fraction abundance is in same order of magnitude as the values reported in Table 3 of Decressin et al. (2007b) which falls well within the considered temperature range. Here also we didn t find any anti-correlation between Mg and Al which is a similar kind of result obtained in (Decressin et al. (2007b)). However we didn t investigate the reason for such observation. Decressin et al. (2007b) has reported that 24 Mg(p,γ) 25 Al reaction rate increment could possibly lead

13 p-capture reaction cycles (I) 13 Table 8 The estimated abundance ratios of [O/Fe],[Na/Fe] and [Al/Fe] for GC M13. a = (Cohen et al. (2005)), b = this work. [ star Fe ] a [ O ] a [ Na ] a [ Al a [ O ] b [ Na ] b [ Al ] b H Fe Fe Fe] Fe Fe Fe II ± ± ± IV ± ± ± II ± ± III ± ± ± K ± ± III ± ± I ± ± I ± ± J ± ± C ± ± II ± ± IV ± ± J ± ± I ± C ± ± C ± ± C ± ± C ± ± C ± ± C ± ± C ± ± C ± ± C ± ± C ± ± C ± ± to such observation. Another consequence is the production of an important amount of primary N 14. Our calculation yields the highest production of this element with mass fraction (Figure1) which is in the same order of magnitude in a stellar model with mass 60M, Z= and initial rotational vel. 800 kms 1 (Decressin et al. (2007b)). We also observe that the O 16 mass fraction estimated in our calculation has the same order of magnitude as that of the mean mass fraction in winds reported in Table5of (Decressin et al. (2007b)) at the end of the central H burning. We also find that the Na 23 has the same order of magnitude (Table5),when compared with Table 5 of (Decressin et al. (2007b)) but these orders are closer to the mass fraction obtained if one uses the high reaction rate constants in the reaction networks. As far as Al 27 is concerned we find that at temperatures greater than T 9 = 0.05 our estimated orders are almost 10 times greater than those estimated by (Decressin et al. (2007b)). The major uncertainties present are the poorly known influence of molecular weight gradients on the shear instabilities and our ignorance of the effect of magnetic fields in the stellar interior on the transport of chemical and angular momentum which are still being considered as prime candidates for affecting the abundance pattern. A star rotating axially, at low metallicity triggers many instabilities in stellar interiors. These instabilities participate in the transport of chemical species and of angular momentum. Amongst the instabilities the shear mixing and the horizontal turbulence is the important agents for effective transport of chemical elements and angular momentum since in a low metallicity star the rotational mixing plays a dominant role (Meynet (2008)). Moreover we have considered the cyclic behaviour of reaction networks, i.e., the evolution of elements is confined. However there may also be some leakage of F 19 going to Ne 20 via a proton-capture, Na 23 (p,γ)mg 24 (Cavallo et al. (1998)) and finally Al 27 takes a proton resulting in the formation of Si 28. This will definitely alter the abundance profile of the considered elements.the possibilities of other internal pollution mechanisms (D Orazi et al. (2013)) within the cluster are also likely to affect the abundance pattern. Theoretical uncertainties are also likely to remain especially in the calculation of reaction rates (Iliadis et al. (2010)), and the choice of the initial

14 14 Upakul Mahanta et al. Table 9 The estimated abundance ratios of [O/Fe],[Na/Fe] and [Al/Fe] for NGC a = (Yong et al. (2005)), b = this work. [ star Fe ] a [ O ] a [ Na ] a [ Al a [ O ] b [ Na ] b [ Al ] b H Fe Fe Fe] Fe Fe Fe NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC6752-mg NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC condition. Also then the mass loss in rotating stars is an asymmetric phenomenon. Very hot stars have dominant polar ejection whereas stars with temperature < K ejects most of the material through equatorial ejection forming a disk which removes a lot of mass of the star as compared to the polar ejection even though mass loss doesn t play a dominant role in low metallicity stars (Meynet (2008)). Nevertheless the elements formed following proton-capture synthesis may have been brought to the surface which carry angular momentum with them and which is why the stars get their rotational motion. Once the rotational speed reaches the critical value the stars start to lose their mass in the form of these synthesized materials in the outer space thus enriching the space by these materials from which the second generation stars are formed. Thus, as the computed abundances of the elements in the present set of initial conditions do not deviate much from the observed values of [O/Fe],[Na/Fe] and [Al/Fe] ratios are concerned, and hence, supports the possibility of occurrence of these nuclear cycles in such kind of a stellar situation at high temperature and low density conditions and thus supporting the primordial scenario. Acknowledgements The author wishes to thank Prof. M.P.Bora and Dr. Bhim Prasad Sharma for their help while completing the manuscript.

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